# American Institute of Mathematical Sciences

## Journals

JIMO
In this paper, we discuss a system of differential equations based on the projection operator for solving the box constrained variational inequality problems. The equilibrium solutions to the differential equation system are proved to be the solutions of the box constrained variational inequality problems. Two differential inclusion problems associated with the system of differential equations are introduced. It is proved that the equilibrium solution to the differential equation system is locally asymptotically stable by verifying the locally asymptotical stability of the equilibrium positions of the differential inclusion problems. An Euler discrete scheme with Armijo line search rule is introduced and its global convergence is demonstrated. The numerical experiments are reported to show that the Euler method is effective.
keywords: global convergence. differential inclusion asymptotical stability differential equations Box constrained variational inequality problem
DCDS
In this paper, we consider the following problem $$\left\{ \begin{array}{ll} -\Delta u+u=u^{2^{*}-1}+\lambda(f(x,u)+h(x))\ \ \hbox{in}\ \mathbb{R}^{N},\\ u\in H^{1}(\mathbb{R}^{N}),\ \ u>0 \ \hbox{in}\ \mathbb{R}^{N}, \end{array} \right. (\star)$$ where $\lambda>0$ is a parameter, $2^* =\frac {2N}{N-2}$ is the critical Sobolev exponent and $N>4$, $f(x,t)$ and $h(x)$ are some given functions. We prove that there exists $0<\lambda^{*}<+\infty$ such that $(\star)$ has exactly two positive solutions for $\lambda\in(0,\lambda^{*})$ by Barrier method and Mountain Pass Lemma and no positive solutions for $\lambda >\lambda^*$. Moreover, if $\lambda=\lambda^*$, $(\star)$ has a unique solution $(\lambda^{*}, u_{\lambda^{*}})$, which means that $(\lambda^{*}, u_{\lambda^{*}})$ is a turning point in $H^{1}(\mathbb{R}^{N})$ for problem $(\star)$.
keywords: variational methods. critical exponent Multiple solutions
DCDS
In this paper, by an approximating argument, we obtain infinitely many radial solutions for the following elliptic systems with critical Sobolev growth $$\left\lbrace\begin{array}{l} -\Delta u=|u|^{2^*-2}u + \frac{η \alpha}{\alpha+β}|u|^{\alpha-2}u |v|^β + \frac{σ p}{p+q} |u|^{p-2}u|v|^q , \ \ x ∈ B , \\ -\Delta v = |v|^{2^*-2}v + \frac{η β}{\alpha+ β } |u|^{\alpha }|v|^{β-2}v + \frac{σ q}{p+q} |u|^{p}|v|^{q-2}v , \ \ x ∈ B , \\ u = v = 0, \ \ &x \in \partial B, \end{array}\right.$$ where $N > \frac{2(p + q + 1) }{p + q - 1}, η, σ > 0, \alpha,β > 1$ and $\alpha + β = 2^* = : \frac{2N}{N-2} ,$ $p,\,q\ge 1$, $2\le p +q<2^*$ and $B\subset \mathbb{R}^N$ is an open ball centered at the origin.
keywords: elliptic systems Radial solution (PS) condition. critical exponent
DCDS-B
We study a mathematical model for the interaction of HIV infection and CD4$^+$ T cells. Local and global analysis is carried out. Let $N$ be the number of HIV virus produced per actively infected T cell. After identifying a critical number $N_{crit}$, we show that if $N\le N_{crit},$ then the uninfected steady state $P_{0}$ is the only equilibrium in the feasible region, and $P_{0}$ is globally asymptotically stable. Therefore, no HIV infection persists. If $N>N_{crit},$ then the infected steady state $P^$* emerges as the unique equilibrium in the interior of the feasible region, $P_{0}$ becomes unstable and the system is uniformly persistent. Therefore, HIV infection persists. In this case, $P^$* can be either stable or unstable. We show that $P^$* is stable only for $r$ (the proliferation rate of T cells) small or large and unstable for some intermediate values of $r.$ In the latter case, numerical simulations indicate a stable periodic solution exists.
keywords: local and global stability HIV infection dynamics Oscillations.
DCDS-B
We discuss the $\omega$-limit set for the Cauchy problem of the porous medium equation with initial data in some weighted spaces. Exactly, we show that there exists some relationship between the $\omega$-limit set of the rescaled initial data and the $\omega$-limit set of the spatially rescaled version of solutions. We also give some applications of such a relationship.
keywords: weighted space. $\omega$-limit set Porous medium equation
DCDS-B
In this paper, a general viral model with virus-driven proliferation of target cells is studied. Global stability results are established by employing the Lyapunov method and a geometric approach developed by Li and Muldowney. It is shown that under certain conditions, the model exhibits a global threshold dynamics, while if these conditions are not met, then backward bifurcation and bistability are possible. An example is presented to provide some insights on how the virus-driven proliferation of target cells influences the virus dynamics and the drug therapy strategies.
keywords: global stability Virus dynamics backward bifurcation. in-host model
DCDS-B
Shadow systems are often used to approximate reaction-diffusion systems when one of the diffusion rates is large. In this paper, we investigate in a shadow system the effects of migration and interspecific competition coefficients on the existence of positive solutions. Our study shows that for any given migration, if the interspecific competition coefficient of the invader is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of a two-species Lotka-Volterra competition-diffusion model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.
keywords: Coexistence state. Lotka-Volterra model Shadow system
DCDS
In the paper we prove the multiplicity existence of both nonlinear Schrödinger equation and Schrödinger system with slow decaying rate of electric potential at infinity. Namely, for any
 $\mathit{\boldsymbol{m}},\mathit{\boldsymbol{n > }}{\bf{0}}$
, the potentials
 $P, Q$
have the asymptotic behavior
 $\left\{ \begin{array}{l}P(r) = 1 + \frac{a}{{{r^m}}} + O\left( {\frac{1}{{{r^{m + \theta }}}}} \right),{\rm{ }}\;\;\;\;\theta > 0\\Q(r) = 1 + \frac{b}{{{r^n}}} + O\left( {\frac{1}{{{r^{n + \widetilde \theta }}}}} \right),\;\;\;\;\;\widetilde \theta > 0\end{array} \right.$
then Schrödinger equation and Schrödinger system have infinitely many solutions with arbitrarily large energy, which extends the results of [37] for single Schrödinger equation and [30] for Schrödinger system.
keywords: Slow decaying nonlinear Schrödinger equations finite dimension reduction
CPAA
We study the dynamics of the one dimensional Swift-Hohenberg equation defined on a large interval $(-l,l)$ with Dirichlet-Neumann boundary conditions, where $l>0$ is large and lies outside of some small neighborhoods of the points $n\pi$ and $(n+1/2)\pi,n \in N$. The arguments are based on dynamical system formulation and bifurcation theory. We show that the system with Dirichlet-Neumann boundary conditions can be reduced to a two-dimensional center manifold for each bifurcation parameter $O(l^{-2})$-close to its critical values when $l$ is sufficiently large. On this invariant manifold, we find families of steady solutions and heteroclinic connections with each connecting two different steady solutions. Moreover, by comparing the above dynamics with that of the Swift-Hohenberg equation defined on $R$ and admitting $2\pi$-spatially periodic solutions in [4], we find that the dynamics in our case preserves the main features of the dynamics in the $2\pi$ spatially periodic case.
keywords: Dirichlet-Neumann boundary. center manifold theory Swift-Hohenberg equation dynamics
PROC